Improved Fourier restriction estimates in higher dimensions

TL;DR

This paper advances Fourier restriction estimates in higher dimensions by refining Guth's polynomial partitioning method with new geometric insights, leading to improved bounds and implications for the Kakeya conjecture.

Contribution

It introduces polynomial Wolff axioms into Guth's recursive approach, achieving better bounds for the restriction problem in high dimensions.

Findings

Improved bounds for the Fourier restriction conjecture in high dimensions.

Enhanced understanding of the Kakeya conjecture implications.

Development of a recursive algorithm incorporating polynomial Wolff axioms.

Abstract

We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.